Algebra and analysis

Presentation

Programme (detailed contents):

Analysis

v  Number series, sequences and series of functions, power series,

v  Normed vector spaces.

Algebra

v  Block diagonalization of matrices,

v  Euclidian and hermitian spaces,

Organisation:

Analysis and Algebra lectures take place in parallel throughout the 1st semester to highlight the links between 2 subjects.

Main difficulties for students:

v  Students confuse sequences and series, linear applications and symmetric bilinear forms. They have difficulty distinguishing the different kinds of convergence, and tend to manipulate vectors as real numbers. They also encounter difficulties in understanding that a single matrix can represent very different objects.

Objectives

At the end of this module, the student will have understood and be able to :

- Determine the different kinds of convergence for number series. Series of functions, power series.
- Find block diagonalization matrix of an endomorphism, manipulate scalar products
and understand the notion of orthogonality in euclidean spaces. Express quadratic forms
as sums of squares.

Needed prerequisite

Analysis and algebra lectures of first year (UF1 et UF2 de mathématiques : I1ANMT11 et 21).

Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...

Bibliography

F.Pécastaings, J. Sevin, Chemin vers l’analyse TOME 2, Vuibert, 1985,ISBN : 2-7117-2174-4.

J.Rivaud,Analyse,Séries ;équations différentielles,Vuibert,1973,ISBN : 2-7117-2133-7.

J. Grifone, Algèbre linéaire,Cépaduès-Edition, 2002, ISBN : 2-85428-239-6

F.Pécastaings,Chemin vers l’algèbre TOME 2, Vuibert, 1986,ISBN : 2-7117-2187-6.